On the Cartan matrix of Mackey algebras

Let k be a field of characteristic p > 0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra μk(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra coμk(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of coμk(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block coμk(b) of coμk(G) is non singular if and only if b is nilpotent with cyclic defect groups. AMS Subject Classification : 18G05, 20C20, 20J06.